Journal of Control and Systems Engineering
Systems Theory Approach to Stabilization of Multiport Networks 



Abstract: 

A theory of stable interconnection of multiport active networks is developed using the stable coprime fractional representation of hybrid network functions at their ports. One of the difficulties in designing stable port interconnection of networks has been that the feedback signal flow graph of the interconnection cannot be easily obtained apart from simplest single port networks. This paper resolves this problem by directly developing the theory in terms of network function of port interconnection. The resulting mathematical problem turns out to have close analogy with stabilization problem of multi input multi output (MIMO) feedback system theory with the major difference that the signal flow graph of a feedback system is never used and that the definition of stability is also not formulated as two input two output map stability as in feedback theory. Hence this problem turns out to be a new application of the coprime fractional approach well known in feedback control. Despite these differences, the conditions for stability turn out to have a close mathematical analogy with the Bezout identity in terms of fractional parts of hybrid network functions analogous to that in feedback theory. A complete parameterization of stabilizing multiport network functions for a given analog multiport network and that of the interconnection is derived. This theory proves that interconnection of physical systems such as networks need not be designed by transforming to feedback signal flow graph and can be handled directly in terms of network interconnections. In particular the stability condition gives a Nyquist stability analog for stability of multiport interconnection without computing an open loop transfer function. The parameterization of the interconnection also facilitates an affine optimization problem for design of such interconnection. Much of the classical theory of synthesis of circuits and their connections is valid only for passive circuits which are always stable. On the other hand active networks can be unstable and can lead to unstable interconnection even if the component parts are stable. Hence this stabilization theory is essential for design of active network interconnections. 

Keywords:Coprime Factorization; Feedback Stabilization; Multiport Network Connections 

Author: Mayuresh Bakshi,Virendra Sule,Maryam Shojaei Baghini 

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